Theorem 3bitr2rd 549

Description: Deduction from transitivity of biconditional.

Hypotheses

Ref	Expression
3bitr2d.1	|- (ph -> (ps <-> ch))
3bitr2d.2	|- (ph -> (th <-> ch))
3bitr2d.3	|- (ph -> (th <-> ta))

Assertion

Ref	Expression
3bitr2rd	|- (ph -> (ta <-> ps))

Proof of Theorem 3bitr2rd

Step	Hyp	Ref	Expression
1		3bitr2d.1	. . 3 |- (ph -> (ps <-> ch))
2		3bitr2d.2	. . 3 |- (ph -> (th <-> ch))
3	1, 2	bitr4d 533	. 2 |- (ph -> (ps <-> th))
4		3bitr2d.3	. 2 |- (ph -> (th <-> ta))
5	3, 4	bitr2d 531	1 |- (ph -> (ta <-> ps))

Syntax hints:   -> wi 3   <-> wb 146

This theorem is referenced by:  muleqaddt 5712  om2uzlt 6299  metgt0 7817  metcn 7886  hvmulcan2t 8935  pjnorm2t 9667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain